I know each element of the central dogma is a code in the mathematical sense of finite-state machines. That is, DNA is transcribed into RNA, and RNA into proteins. What other examples are there of codes that exist naturally in organisms?
DNA is a code that occurs via stepwise polymerization, like all major macromolecules. In respects to DNA, it is Nucleic Acid sequences that polymerize to create the DNA. In eukaryotic cells this happens at multiple replicons (regions of replication). So you can look DNA as code for itself. You also have rRNA, mRNA, and tRNA. All of these code for and do specific things. rRNA facilitates the creation and movement of the ribosome, where mRNA and tRNA "code" for the final protein. tRNA serves as the principle code strand for the correct Amino Acids. This is read as a series of three N.A's (codons) One codon for a respective Amino Acid. mRNA actually guides the synthesis of these polypeptide chains. You also have something called operons. These are groups of genes with related functions that are clustered together. This allows for rapid initiation and termination of gene expression. Im sure there are a few others but these are those that come off the top of my head. The term "code" and molecular bio are very much intertwined.
I am not very familiar with them, but I hear that bacterial non-ribosomal peptide synthases are made of catalytic modules, lined up in series, paralleling their product. My suspicion is that there is some promiscuity / ambiguity in the nonribosomal code.
Also, N-glycosylation in the ER / Golgi occurs in a defined series of events, but that includes backtracking (that is, sugar cleavage), which I am not sure would belong in a finite state machine.